Spectral enclosures for a class of block operator matrices filespectral enclosures for a class of block operator matrices juan giribet, matthias langer, francisco mart´inez per ´ia,

Technische Universität Ilmenau Institut für Mathematik

Preprint No. M 19/04

Spectral enclosures for a class of block operator matrices

Juan Giribet, Matthias Langer, Francisco Martinez Peria, Friedrich Philipp and Carsten Trunk

März 2019

Impressum: Hrsg.: Leiter des Instituts für Mathematik

Weimarer Straße 25 98693 Ilmenau

Tel.: +49 3677 69-3621 Fax: +49 3677 69-3270 http://www.tu-ilmenau.de/math/

URN: urn:nbn:de:gbv:ilm1-2019200198

SPECTRAL ENCLOSURES FOR A CLASS OF BLOCK OPERATOR MATRICES

JUAN GIRIBET, MATTHIAS LANGER, FRANCISCO MARTINEZ PERIA, FRIEDRICH PHILIPP,AND CARSTEN TRUNK

ABSTRACT. We prove new spectral enclosures for the non-real spectrum of a class of 2×2 blockoperator matrices with self-adjoint operators A and D on the diagonal and operators B and −B∗

as off-diagonal entries. One of our main results resembles Gershgorin’s circle theorem. Theenclosures are applied to J-frame operators.

1. Introduction

We consider block operator matrices S acting in the orthogonal sum H := H+⊕H− of twoHilbert spaces,

S =

[A B−B∗ D

], (1.1)

where A and D are (possibly unbounded) self-adjoint operators in H+ and H−, respectively, andB is a bounded operator from H− to H+.

Such operators play an important role in various applications. For instance, they appear in thestudy of so called floating singularities [6, 13, 14, 18, 20], in the perturbation theory for equationsof indefinite Sturm–Liouville type [5], and also in frame theory [11, 12].

Of particular interest is the location of the spectrum of S. In [17, 22, 27] spectral enclosureswere obtained via the quadratic numerical range, and in [4, 5] in terms of the spectra of A and D.Gershgorin-type results for more general operator matrices were presented in [8] and [25]. More-over, in [1, 13, 19, 20] the essential spectrum was investigated and in [18] variational principlesand estimates for eigenvalues were proved. Invariant subspaces and factorizations of Schur com-plements were considered in [23] and [2], and in [3] conditions were presented for an operatorof the form (1.1) to be similar to a self-adjoint operator in a Hilbert space. For an overview werefer to the monograph [26].

In general, the spectrum of block operator matrices as (1.1) is not necessarily contained in thereal line. The aim of this paper is to provide enclosures for the (non-real) spectrum of S in termsof (spectral) quantities of the operators A, B, and D.

We start with a general enclosure for the (closure) of the quadratic numerical range of S,formulated in terms of the numerical ranges of A and D and the norm of B; see Theorem 3.1below. The quadratic numerical range of a block operator matrix was introduced in [21] and itsclosure contains the spectrum of S; see (2.2). Although similar enclosures for the spectrum ofS were already known, one of the advantages of having a spectral enclosure for the quadraticnumerical range is that it leads also to estimates of the norm of the resolvent; see the discussionin Remark 3.2. Moreover, Theorem 3.1 is sharp in the sense that the enclosures for the quadratic

2010 Mathematics Subject Classification. Primary 47A10; Secondary 47B50, 47A12, 81Q15.Key words and phrases. Block operator matrices; quadratic numerical range; spectral enclosure; Gershgorin’s circle.F. Martınez Perıa, and C. Trunk gratefully acknowledge the support of the DFG (Deutsche Forschungsgemeinschaft)from the project TR 903/21-1. In addition, J. I. Giribet and F. Martınez Perıa gratefully acknowledges the supportfrom the grant PIP CONICET 0168.

1

2 J. GIRIBET, M. LANGER, F. MARTINEZ PERIA, F. PHILIPP, AND C. TRUNK

numerical range cannot be improved if just the numerical ranges of A and D and the norm of Bare known; see Proposition 3.3.

The main contribution of this paper is a spectral enclosure for the operator matrix S, which isconnected with the Schur complements. It is well known and follows from a relatively simpleNeumann series type argument applied to the first and second Schur complement that

σ(S)\R ⊆{

λ ∈ C\R : ‖B∗(A−λ )−1B(D−λ )−1‖ ≥ 1 and ‖B(D−λ )−1B∗(A−λ )−1‖ ≥ 1}

;(1.2)

see [8, Theorem 1.1], [4, Lemma 5.2 (ii)] or [26, Section 2.3]. Here we prove that

σ(S)\R ⊆{

λ ∈ C\R : ‖(A−λ )−1B‖ ≥ 1 and ‖(D−λ )−1B∗‖ ≥ 1}

; (1.3)

see Theorem 4.3 below. The enclosures (1.2) and (1.3) are independent of each other in the sensethat none of the sets in the right-hand sides of (1.2) and (1.3) is strictly contained in the otherone. However, since the norm inequalities in (1.3) deal separately with the resolvent functions ofA and D it is easy to construct examples where the enclosure in (1.3) is strictly contained in theone in (1.2), see Example 4.9.

Note that the spectral enclosures in (1.2) and (1.3) are not explicitly formulated in terms of thespectra of A and D. However, it is one of our main observations that (1.3) allows a reformulationin a more geometric manner. In particular, given λ ∈ C\R, ‖(A−λ )−1B‖ ≥ 1 if and only if forall positive continuous functions f : R→ R with some specific behaviour at infinity we have

λ ∈⋃

t∈σB(A)

B f (t)−1‖ f (A)B‖(t),

where σB(A) is a specific closed subset of σ(A) and Br(t) stands for the closed ball of radius raround t; for details we refer to Proposition 4.7 below. A similar interpretation can be obtainedfor the inequality ‖(D−λ )−1B∗‖ ≥ 1 in terms of continuous functions defined in a closed subsetσB∗(D) of the spectrum of D. Therefore, the enclosure for the non-real part of the spectrum of Sin (1.3) implies a family of enclosures which resemble Gershgorin’s circle theorem: for any twopositive continuous functions f and g with some specific behaviour at infinity we have that

σ(S)\R ⊆

⋃t∈σB(A)

B f (t)−1‖ f (A)B‖(t)

∩ ⋃

s∈σB∗ (D)

Bg(s)−1‖g(D)B∗‖(s)

; (1.4)

see Theorem 4.8 below.Maybe the most interesting situations appear when one chooses the functions f and g explic-

itly. For instance, if A is boundedly invertible and f (t) = |t|−1 then (1.4) implies

σ(S)\R⊆⋃

a∈σB(A)

B|a|‖A−1B‖(a). (1.5)

Moreover, if ‖A−1B‖ < 1 then these balls are contained in a double-sector with half openingangle arcsin‖A−1B‖, see Figure 1 below.

It is worth mentioning that (1.3) also improves the following spectral enclosure obtained in[5]: if Br(∆) = {z ∈ C : dist(z,∆)≤ r} then

σ(S)\R ⊆ B‖B‖(σ(A)) ∩ B‖B‖(σ(D)).

In fact, ‖(A−λ )−1B‖ ≥ 1 obviously implies ‖(A−λ )−1‖‖B‖ ≥ 1 and the latter is equivalent toλ ∈ B‖B‖(σ(A)). A similar argument with D and B∗ instead of A and B completes the proof.

Finally, in Section 5 we apply the spectral enclosures obtained in Section 4 to operator matricesof the form (1.1) which appear in frame theory. The so-called J-frame operators were introduced

SPECTRAL ENCLOSURES FOR A CLASS OF BLOCK OPERATOR MATRICES 3

FIGURE 1. The spectral enclosure for σ(S) \R given in (1.5) (in orange) andthe set σB(A) (in blue).

in [12] and further investigated in [11]. Our findings lead to significant improvements of thespectral enclosures for J-frame operators obtained in [11].

2. Preliminaries

If H and K are Hilbert spaces, we denote by L(H ,K ) the space of all bounded linear oper-ators mapping from H to K . As usual, we set L(H ) := L(H ,H ). For r ≥ 0 and ∆ ⊆ C weset

Br(∆) := {z ∈ C : dist(z,∆)≤ r}.If a ∈ C, we also write Br(a) := Br({a}) for the closed disc with centre a and radius r.

The numerical range of a linear operator T in the Hilbert space H is defined by

W (T ) := {(T x,x) : x ∈ dom T, ‖x‖= 1}.

It is well known that the numerical range W (T ) is convex and that C \W (T ) has at most two(open) connected components (see [15, V.3.2]). Moreover, it is immediate from the definitionof W (T ) that σp(T ) ⊆W (T ), where σp(T ) stands for the point spectrum of T . If T is closed,λ ∈ C\W (T ) and x ∈ dom T , ‖x‖= 1, then

‖(T −λ )x‖ ≥ |(T x,x)−λ | ≥ dist(λ ,W (T )).

This shows that ran(T −λ ) is closed and ker(T −λ ) = {0}. Hence, if each of the (at most two)components of C\W (T ) contains points from the resolvent set ρ(T ), then σ(T )⊆W (T ). Thisholds in particular if T is bounded. The next lemma is now immediate.

Lemma 2.1. Let T = A+B, where A is a self-adjoint operator in a Hilbert space H and B ∈L(H ). Then σ(T )⊆W (T ) and for λ /∈W (T ) we have

‖(T −λ )−1‖ ≤ 1

dist(λ ,W (T )).

Let us also recall the definition of the quadratic numerical range, which was introduced in[21]; see also [26, Definition 1.1.1]. Assume that the Hilbert space H is the orthogonal sum oftwo Hilbert spaces, H+ and H−. Let S be a bounded operator in H decomposed as

S =

[A BC D

],


where A∈ L(H+), B∈ L(H−,H+), C ∈ L(H+,H−), and D∈ L(H−). For f ∈H+ and g∈H−with ‖ f‖= ‖g‖= 1 we introduce the 2×2 matrix

S f ,g =

[(A f , f ) (Bg, f )

(C f ,g) (Dg,g)

]. (2.1)

The setW 2(S) :=

⋃f∈H+,g∈H−‖ f‖=‖g‖=1

σp(S f ,g)

is called the quadratic numerical range of S. It is no longer a convex subset of C, but it has atmost two connected components.

One of the advantages of the quadratic numerical range is that it is contained in the numericalrange: W 2(S)⊆W (S) and that we have the following refined spectral inclusions

σp(S)⊆W 2(S) and σ(S)⊆W 2(S); (2.2)

see [26, Theorem 1.3.1]. Moreover, the resolvent can be estimated in terms of the distance toW 2(S):

‖(S−λ )−1‖ ≤ ‖S‖+ |λ |[dist(λ ,W 2(S))]2

, λ /∈W 2(S); (2.3)

see [26, Theorem 1.4.1]. If W 2(S) = F1∪F2 with disjoint non-empty closed sets F1 and F2, then

‖(S−λ )−1‖ ≤ ‖S‖+ |λ |dist(λ ,F1)dist(λ ,F2)

, λ /∈W 2(S); (2.4)

see [26, Theorem 1.4.5].The quadratic numerical range definition can be easily extended to unbounded block operator

matrices, restricting the vectors f and g in (2.1) to the proper domains (dom A)∩ (dom C) and(dom B)∩ (dom D), respectively. For details see [26, Definition 2.5.1].

In the following we recall the definition of the Schur complements of a block operator matrix,which are powerful tools to study the spectrum and spectral properties. Let A and D be closedoperators in H+ and H−, respectively, B ∈ L(H−,H+), and C ∈ L(H+,H−). For the blockoperator matrix

S =

[A BC D

], dom S = dom A⊕dom D,

the first and second Schur complements of S are defined by:

S1(λ ) := A−λ −B(D−λ )−1C, λ ∈ ρ(D), (2.5)

S2(λ ) := D−λ −C(A−λ )−1B, λ ∈ ρ(A). (2.6)

These are analytic operator functions defined on the resolvent sets of D and A, respectively.In the next sections we shall make use of the following (simplified) auxiliary result from [8];

see also [26, Theorem 2.3.3].

Lemma 2.2 ([8, Lemma 2.1]). Let A and D be closed operators in H+ and H−, respectively, letB ∈ L(H−,H+) and C ∈ L(H+,H−), and consider the block operator matrix

S =

[A BC D

], dom S = dom A⊕dom D.

Then the following statements hold:

(i) For λ ∈ ρ(D) one has λ ∈ σ(S) if and only if 0 ∈ σ(S1(λ )).


(ii) For λ ∈ ρ(A) one has λ ∈ σ(S) if and only if 0 ∈ σ(S2(λ )).

Moreover, if λ ∈ ρ(D)∩ρ(S), then

(S−λ )−1 =

[I 0

−(D−λ )−1C I

][S1(λ )

−1 00 (D−λ )−1

][I −B(D−λ )−1

0 I

].

3. Enclosures for the quadratic numerical range

Let S be as in (1.1) with bounded operators

A ∈ L(H+), B ∈ L(H−,H+), D ∈ L(H−),

where A and D are self-adjoint in the Hilbert spaces H+ and H−, respectively. Hence, thenumerical ranges W (A) and W (D) are real intervals. We introduce the following numbers, whichare used for the description of the spectral enclosures that are proved below,

a− := inf W (A), a+ := sup W (A), (3.1)

d− := inf W (D), d+ := sup W (D), (3.2)

m− :=a−+d−

2, m+ :=

a++d+2

, (3.3)

c :=12[min{a+,d+}+max{a−,d−}

], (3.4)

`=12

dist(W (A),W (D)

). (3.5)

If W (A)∩W (D) = ∅, then c is the midpoint of the gap between W (A) = [a−,a+] and W (D) =[d−,d+], and ` is half the length of the gap; e.g. if d+ < a−, then

c = 12(d++a−) and `= 1

2(a−−d+).

The following theorem contains enclosures of the closure of the quadratic numerical range ofoperators of the form (1.1). Due to (2.2) these yield also enclosures for the spectrum. For relatedresults see Remark 3.2 below. Note that the enclosures in Theorem 3.1 depend only on W (A),W (D) and ‖B‖. These enclosures are illustrated in Figures 2–5 below.

Theorem 3.1. Given a Hilbert space H = H+⊕H−, consider the block operator matrix

S :=[

A B−B∗ D

](3.6)

where B∈ L(H−,H+), and A and D are bounded self-adjoint operators in H+ and H−, respec-tively. Further, let the constants a±, d±, m±, c and ` be as in (3.1)–(3.3). Then

W 2(S)∩R⊆[min{a−,d−},max{a+,d+}

], (3.7)

W 2(S)\R⊆ B‖B‖([a−,a+]

)∩ B‖B‖

([d−,d+]

)∩ {z ∈ C : m− ≤ Re z≤ m+}. (3.8)

Moreover, in the special case where

W (A)∩W (D) =∅ and ‖B‖ ≤ `, (3.9)

we have

W 2(S)⊆

[min{a−,d−}, c−

√`2−‖B‖2

]∪

[c+√`2−‖B‖2, max{a+,d+}

]; (3.10)

in particular, the quadratic numerical range and the spectrum of S are real.


Proof. Since the right-hand sides of (3.7), (3.8), and (3.10) are closed, it is sufficient to provethat W 2(S)∩R, W 2(S) \R, and W 2(S) are contained in the right-hand sides of (3.7), (3.8), and(3.10), respectively.

Let z ∈W 2(S). Then there exist f ∈H+ and g ∈H− with ‖ f‖ = ‖g‖ = 1 such that z is aneigenvalue of the matrix S f ,g in (2.1). Set

α := (A f , f ), β := (Bg, f ), δ := (Dg,g).

Then

α ∈W (A)⊆ [a−,a+], δ ∈W (D)⊆ [d−,d+], |β | ≤ ‖B‖

and z = z+ or z = z−, where

z± :=α +δ

2±√(

α−δ

2

)2−|β |2

and we use the convention that√

w≥ 0 if w ∈ [0,∞) and Im√

w > 0 if w ∈ (−∞,0).Let us first consider the case when z = x+ iy /∈ R. Since all involved sets are symmetric with

respect to the real axis, we can assume, without loss of generality, that y > 0; hence

x =α +δ

2, y =

√|β |2−

(α−δ

2

)2.

Clearly, m− ≤ x ≤ m+, which shows that z is contained in the last set on the right-hand side of(3.8). Further, we have

(x−α)2 + y2 =(−α +δ

2

)2+ |β |2−

(α−δ

2

)2= |β |2 ≤ ‖B‖2,

which yields that

dist(z, [a−,a+])≤ |z−α| ≤ ‖B‖.

In a similar way one shows that dist(z, [d−,d+]) ≤ ‖B‖, which proves that z is contained in theright-hand side of (3.8).

Now assume that z ∈ R. Then

z≤ z+ ≤α +δ

2+

∣∣∣∣α−δ

2

∣∣∣∣= max{α,δ} ≤max{a+,d+}.

Similarly, one shows that z ≥ min{a−,d−}, which proves that z is contained in the right-handside of (3.7).

Finally, assume that (3.9) is satisfied. Since in this case B‖B‖([a−,a+])∩B‖B‖([d−,d+]) = ∅,the already proved relation (3.8) implies that W 2(S)⊆R. To show (3.10) we assume, without lossof generality, that d+ < a−, in which case c= 1

2(a−+d+), `= 12(a−−d+) and δ ≤ d+ < a−≤α .

It is easy to see that z+ is increasing in α and decreasing in δ whenever it is real and α > δ . Hence

z+ ≥a−+d+

2+

√(a−−d+2

)2−|β |2 = c+

√`2−‖B‖2

and similarly,

z− ≤a−+d+

2−√(a−−d+

2

)2−|β |2 = c−

√`2−‖B‖2 .

Together with (3.7), this shows the inclusion (3.10). �


Remark 3.2. (a) The following inclusions for the spectrum

σ(S)∩R⊆[min{a−,d−},max{a+,d+}

]and σ(S)\R⊆ {z ∈ C : m− ≤ Re z≤ m+}

were proved in [17, Theorem 2.1]. Moreover, it was proved in [5, Theorem 3.5] (cf. also [19,Theorem 4.2]) that

σ(S)\R⊆ B‖B‖(σ(A)

)∩ B‖B‖

(σ(D)

), (3.11)

which implies that σ(S)\R ⊆ B‖B‖([a−,a+])∩B‖B‖([d−,d+]). We point out that, although (3.11)gives (in general) a sharper estimate for the spectrum, the advantage of also having the enclosuresfor the quadratic numerical range is that the resolvent estimates (2.3) and (2.4) can be applied.

(b) It follows easily from (3.8) that for the case when ‖B‖> ` we have

σ(S)⊆W 2(S)⊆{

z ∈ C : | Im z| ≤√‖B‖2− `2

};

cf. [27, Proposition 1.3.9] or [27, Theorem 5.5] for this particular enclosure of the spectrum.

(c) Note also that, for the spectrum, the enclosure (3.7) can be derived from [3, Theorem 5.8]or [4, Theorem 5.4]. For enclosures for the quadratic numerical range and the spectrum where allentries A, B, and D are allowed to be unbounded see [22, Proposition 4.10 and Theorem 4.13].

Figures 2–5 below show the enclosures for W 2(S) from Theorem 3.1. In Figures 2 and 3 thesituation where W (A)∩W (D) 6=∅ is considered. If ‖B‖ is less than or equal to

τ := min{

m+−min{a+,d+}, max{a−,d−}−m−},

the non-real spectrum is contained in B‖B‖([a−,a+])∩ B‖B‖([d−,d+]) = B‖B‖(W (A)∩W (D)).When ‖B‖ > τ then the enclosure in {z ∈ C : m− ≤ Re z ≤ m+}, the third set on the right-handside of (3.8), has to be taken into account as well.

The case when there is a gap between W (A) and W (D) is considered in Figures 4 and 5. When‖B‖ is small, then W 2(S) is contained in the union of the two real intervals on the right-hand sideof (3.10). When ‖B‖ is larger, then the spectrum may be non-real, and the right-hand sides of(3.7) and (3.8) have to be used.

d− m− a− d+ m+ a+

FIGURE 2. The region (indicated in orange) given by the union of the sets on theright-hand sides of (3.7) and (3.8) that contains W 2(S), when W (A) and W (D)overlap and ‖B‖< τ .


d− m− a− d+ m+ a+

FIGURE 3. The region (indicated in orange) given by the union of the sets on theright-hand sides of (3.7) and (3.8) that contains W 2(S), when W (A) and W (D)overlap and ‖B‖> τ .

d− d+ c a− a+

FIGURE 4. The two intervals (indicated in orange) on the right-hand side of(3.10) whose union contains W 2(S), when W (A) and W (D) are separated and‖B‖< τ .

d− d+ m− c m+a− a+

FIGURE 5. The region (indicated in orange) given by the union of the sets on theright-hand sides of (3.7) and (3.8) that contains W 2(S), when W (A) and W (D)are separated and ‖B‖> τ .

The next proposition shows that Theorem 3.1 is sharp in the sense that given [a−,a+] =W (A),[d−,d+] =W (D) and ‖B‖, the enclosures for the spectrum and the quadratic numerical range ofS cannot be improved, i.e. an operator S is constructed for which equality holds in (3.7) and (3.8),and in (3.10) if (3.9) is satisfied.


Proposition 3.3. Let a+,a−,d+,d−,b ∈ R such that a− ≤ a+, d− ≤ d+, and b > 0. Then thereexist separable Hilbert spaces H±, self-adjoint operators A and D in H+ and H−, respectively,and B ∈ L(H−,H+) such that

W (A) = [a−,a+], W (D) = [d−,d+], ‖B‖= b,

and (with the notation from (3.3)–(3.5)) the operator S from (3.6) satisfies

σ(S) =W 2(S) =(

Bb([a−,a+]

)∩Bb

([d−,d+]

)∩{z ∈ C : m− ≤ Re z≤ m+}

)∪[min{a−,d−},max{a+,d+}

] (3.12)

if b > ` and

σ(S) =W 2(S) =

[min{a−,d−}, c−

√`2

4−b2

]∪

[c+

√`2

4−b2, max{a+,d+}

](3.13)

if b≤ `.

Proof. Let H+ = H− = `2 and define the operators

A = diag(a1,a2, . . .), B = diag(b1,b2, . . .), D = diag(d1,d2, . . .)

with numbersan ∈ [a−,a+], bn ∈ [0,b], dn ∈ [d−,d+], (3.14)

n ∈N, which are chosen later. Let zn = xn+ iyn, n ∈N, be such that {zn : n ∈N} is a dense subsetof the right-hand sides of (3.12) or (3.13), respectively. Below we construct an,bn,dn such that

zn = w+(an,bn,dn) or zn = w−(an,bn,dn), (3.15)

where

w±(an,bn,dn) :=an +dn

2±√(an−dn

2

)2−|bn|2. (3.16)

Since then zn is an eigenvalue of S and σ(S) is closed, this, together with the enclosures inTheorem 3.1, shows equality in (3.12) and (3.13).

Let us first consider the case when zn /∈ R. Then zn is in the right-hand side of (3.12). Ifxn ∈ [a−,a+]∩ [d−,d+], then set

an := dn := xn, bn := |yn|.

Clearly,an ∈ [a−,a+], dn ∈ [d−,d+], bn = |yn|= dist(zn, [a−,a+])≤ b,

and (3.15) is satisfied. Now assume that xn /∈ [a−,a+]∩ [d−,d+]. Without loss of generality wecan assume that

dist(xn, [a−,a+]

)≤ dist

(xn, [d−,d+]

), (3.17)

which implies that xn /∈ [d−,d+]. Let us consider the case when xn < d−; the case xn > d+ isanalogous. Set

an = 2xn−d−, dn = d−, bn =√

y2n +(xn−d−)2.

Clearly, dn ∈ [d−,d+]. From xn ≥ m− we obtain that

an = 2xn−d− ≥ 2m−−d− = 2a−+d−

2−d− = a−.


If xn ≤ a+, then an = xn +(xn− d−) < xn ≤ a+ and hence an ∈ [a−,a+]. If xn > a+, then theinequality in (3.17) is equivalent to xn−a+ ≤ d−− xn, which implies that

an = 2xn−d− ≤ a+;

hence also in this case we have an ∈ [a−,a+]. Moreover,

bn =√

y2n +(xn−d−)2 = |zn−d−|= dist

(zn, [d−,d+]

)≤ b.

It is easy to check that (3.15) is satisfied.Next we consider the case when zn ∈ R. If zn ∈ [a−,a+], then choose an = zn, dn arbitrary in

[d−,d+] and bn = 0. Then

w+(an,bn,dn) = max{zn,dn}, w−(an,bn,dn) = min{zn,dn},and hence (3.15) holds. The case when zn ∈ [d−,d+] is similar. If [a−,a+]∩ [d−,d+] 6= ∅, thenall cases of real zn are covered. Finally, assume that [a−,a+]∩ [d−,d+] = ∅ and zn /∈ [a−,a+]∪[d−,d+]. Without loss of generality we can assume that d+ < a−; then zn ∈ (d+,a−). Let usconsider the case when zn ≥ c = 1

2(d++a−); the other case is similar. Set

an = a−, dn = d+, bn =

√( `2

)2− (zn− c)2.

It is easy to check that zn = w+(an,bn,dn). If b > 12 dist([a−,a+], [d−,d+]) = `

2 , then bn ≤ `2 < b.

If b≤ `2 , then the form of the right-hand side of (3.13) implies that

zn ≥ c+

√( `2

)2−b2,

which yields

bn ≤

√( `2

)2−(( `

2

)2−b2

)= b.

The relations in (3.14) imply that the operators A, B and D are bounded with ‖B‖≤ b. If we had‖B‖ < b, then we would obtain a strictly smaller enclosure for the spectrum from Theorem 3.1,which contradicts the already obtained equality in (3.12) or (3.13), respectively. �

4. Gershgorin-type enclosure for the spectrum of block operator matrices

In this section we provide another spectral enclosure for the non-real spectrum of the blockoperator matrix

S =

[A B−B∗ D

], dom S = dom A⊕dom D. (4.1)

As already indicated by (4.1), we also allow unbounded entries A and D. The operator B remainsbounded in our considerations. The result has similarities with Gershgorin’s circle theorem formatrices [10] and block operator matrices [25, 26, 8] since we show that the non-real spectrum ofthe operator matrix S is contained in the union of a family of closed balls, centred along parts ofthe spectrum of the block A in the diagonal of S (see (4.3)). To formulate the result, for a closedset M ⊆ R define the following class of continuous functions:

C+(M) ={

f ∈C(M) : f (t)> 0 for t ∈M, supt∈M

f (t)< ∞, inf|t|≥1|t| f (t)> 0

}. (4.2)

The last two conditions obviously only matter if M is unbounded. If M is compact, then C+(M)is the set of positive continuous functions on M. Note that any positive constant function iscontained in C+(M) and also |t|−1 ∈C+(M) if 0 /∈M.


Theorem 4.1. Given a Hilbert space H = H+⊕H−, consider the block operator matrix S in(4.1), where B∈ L(H−,H+) and A and D are self-adjoint operators in H+ and H−, respectively.Then, for any f ∈C+(σ(A)) and g ∈C+(σ(D)) we have

σ(S)\R ⊆

⋃t∈σ(A)

B f (t)−1‖ f (A)B‖(t)

∩ ⋃

s∈σ(D)

Bg(s)−1‖g(D)B∗‖(s)

. (4.3)

Remark 4.2. If we set f = g = 1 in Theorem 4.1, then the spectral inclusion (4.3) becomes

σ(S)\R ⊆ B‖B‖(σ(A)) ∩ B‖B‖(σ(D)), (4.4)

which was already proved in [5, Theorem 3.5], cf. (3.11).

Theorem 4.1 will follow from Theorem 4.3 below, which is an improvement of [5, Theo-rem 3.5]. The spectral inclusion (4.4) means that a non-real point in the spectrum of S satisfiesdist(λ ,σ(A)) ≤ ‖B‖ and dist(λ ,σ(D)) ≤ ‖B‖. This is equivalent to ‖(A−λ )−1‖‖B‖ ≥ 1 and‖(D−λ )−1‖‖B∗‖ ≥ 1. Hence, the spectral enclosure given in (4.5) is sharper.

Theorem 4.3. Given a Hilbert space H = H+⊕H−, consider the block operator matrix S in(4.1), where B∈ L(H−,H+) and A and D are self-adjoint operators in H+ and H−, respectively.Then

σ(S)\R ⊆{

λ ∈ C\R : ‖(A−λ )−1B‖ ≥ 1 and ‖(D−λ )−1B∗‖ ≥ 1}. (4.5)

Moreover, given λ ∈ C\R then

‖(S−λ )−1‖ ≤ 1+‖(A−λ )−1B‖+‖(A−λ )−1B‖2

| Im λ | · (1−‖(A−λ )−1B‖2)if ‖(A−λ )−1B‖< 1, (4.6)

‖(S−λ )−1‖ ≤ 1+‖(D−λ )−1B∗‖+‖(D−λ )−1B∗‖2

| Im λ | · (1−‖(D−λ )−1B∗‖2)if ‖(D−λ )−1B∗‖< 1. (4.7)

Before we prove Theorem 4.3, we provide a couple of remarks and an example.

Remark 4.4. (a) The same conclusions as in Theorem 4.3 hold, if we drop the boundednessassumption on B and, instead, assume that B is D-bounded with D-bound less than one and B∗ isA-bounded with A-bound less than one. The arguments in the proof are essentially the same.

(b) Note that both σ(S)\R and the right-hand side of (4.5) are sets which are symmetric withrespect to the real axis.

(c) There is another spectral enclosure for the operator matrix S that results from a relativelysimple argument (see [8, Theorem 1.1] or [4, Lemma 5.2 (ii)]). Consider the second Schur com-plement S2(λ ) = D−λ +B∗(A−λ )−1B for λ ∈ C \R. Applying (D−λ )−1 from the left andfrom the right we obtain

S2(λ )(D−λ )−1 = I +B∗(A−λ )−1B(D−λ )−1 and

(D−λ )−1S2(λ ) = I +(D−λ )−1B∗(A−λ )−1B|dom D.

That is, if one of

NS(λ ) := ‖B∗(A−λ )−1B(D−λ )−1‖ or NS(λ ) = ‖(D−λ )−1B∗(A−λ )−1B‖is less than 1, then the Schur complement S2(λ ) is boundedly invertible and so λ ∈ ρ(S). Hence,

σ(S) ⊆{

λ ∈ ρ(D) : NS(λ )≥ 1 and NS(λ )≥ 1}.


A similar reasoning applies to the first Schur complement S1(λ ) = A−λ +B(D−λ )−1B∗ andgives

σ(S) ⊆{

λ ∈ ρ(A) : MS(λ )≥ 1 and MS(λ )≥ 1},

where MS(λ ) := ‖B(D−λ )−1B∗(A−λ )−1‖. This implies that

σ(S)\R ⊆{

λ ∈ C\R : min{NS(λ ),NS(λ ),MS(λ ),MS(λ )} ≥ 1}. (4.8)

(d) The spectral enclosures (4.5) and (4.8) are independent of each other, meaning that, ingeneral, none of the corresponding sets on the right-hand sides of the two relations contains theother. Consequently, if we intersect the right-hand side of the already known enclosure (4.8)with the new one (4.5), we obtain a better bound for the non-real spectrum of S, as illustrated inExample 4.5 below. However, Example 4.9 shows a situation, where (4.5) is in fact strictly betterthan (4.8).

(e) Both spectral enclosures (4.5) and (4.8) require complete knowledge about the functions‖(A−·)−1B‖, ‖(D−·)−1B∗‖, NS and MS. In contrast, Theorem 4.1 basically only requires knowl-edge about σ(A) and σ(D) and is therefore better suited for computations.

Example 4.5. We let H− = H+ = C2 and

A =

[2 1+ i

1− i −1

], D =

[1 00 −5

], B =

[i 1+ i

2

−1− i − 25

].

The four eigenvalues of S ∈C4×4 are depicted as black dots in the figure below. Note that two ofthem are real. They are (approximately)−4.73166, 2.38898,−0.328657±1.03244 i. The regionfrom (4.8) is bounded by the three red curves, while the two blue curves bound the region on theright-hand side of (4.5). The orange filled region is the intersection of the two enclosures.

FIGURE 6. The spectral enclosures (4.5), bounded with blue curves, and (4.8),bounded with orange curves, for the matrix in Example 4.5.

Proof of Theorem 4.3. We use the first Schur complement S1 of the block operator matrix S in(4.1), which is given by

S1(λ ) = A−λ +B(D−λ )−1B∗, dom S1(λ ) = dom A,

for λ ∈ ρ(D); see (2.5).For λ ∈ C\R we have S1(λ )

∗ = S1(λ ) and, setting T := (D−λ )−1B∗ we obtain

S1(λ )−S1(λ ) = λ −λ +B[(D−λ )−1− (D−λ )−1]B∗ = (λ −λ )(T ∗T − I).

If Im λ > 0, then for arbitrary h ∈ dom A with ‖h‖= 1 we have

Im(S1(λ )h,h) =12i

((S1(λ )−S1(λ ))h,h

)= (Im λ ) · (‖T h‖2−1) ≤ (Im λ )(‖T‖2−1). (4.9)


In particular, if ‖T‖< 1, then 0 /∈W (S1(λ )) and therefore 0 /∈ σ(S1(λ )); see Lemma 2.1. NowLemma 2.2 implies that λ /∈ σ(S). A similar reasoning applies to the case Im λ < 0. This provesthat

σ(S)\R ⊆{

λ ∈ C\R : ‖(D−λ )−1B∗‖ ≥ 1}.

Applying the same arguments to the second Schur complement S2, we obtain

σ(S)\R ⊆{

λ ∈ C\R : ‖(A−λ )−1B‖ ≥ 1},

which completes the proof of the inclusion (4.5).Note also that (4.9) implies that

dist(0,W (S1(λ ))

)≥ | Im λ |(1−‖T‖2). (4.10)

if ‖T‖ < 1. Let us now prove the estimate (4.7) for the resolvent of S. For this, let λ ∈ C \Rsuch that ‖T‖< 1, where T = (D−λ )−1B∗ as above. By Lemma 2.2 we have

(S−λ )−1 =

[I 0

(D−λ )−1B∗ I

][S1(λ )

−1 00 (D−λ )−1

][I −B(D−λ )−1

0 I

]. (4.11)

Denote the first factor by L. Then

‖L‖2 = ‖L∗L‖=∥∥∥∥[ I T ∗

0 I

][I 0T I

]∥∥∥∥= ∥∥∥∥[ I +T ∗T T ∗

T I

]∥∥∥∥≤∥∥∥∥[ I 0

0 I

]∥∥∥∥+∥∥∥∥[ T ∗T 00 0

]∥∥∥∥+∥∥∥∥[ 0 T ∗

T 0

]∥∥∥∥= 1+‖T‖2 +‖T‖.

Since (D−λ )−1 is normal, we have

‖B(D−λ )−1‖= ‖(D−λ )−1B∗‖= ‖(D−λ )−1B∗‖= ‖T‖,which implies that for the last factor in (4.11) we have the same estimate as for the first one. Itremains to estimate the middle factor M in (4.11). To this end, note that Lemma 2.1 and (4.10)yield

‖S1(λ )−1‖ ≤ dist

(0,W (S1(λ ))

)−1 ≤ | Im λ |−1(1−‖T‖2)−1.

Since ‖(D−λ )−1‖ ≤ | Im λ |−1, we obtain

‖M‖= max{‖S1(λ )

−1‖, ‖(D−λ )−1‖}≤ | Im λ |−1(1−‖T‖2)−1.

Hence

‖(S−λ )−1‖ ≤ 1+‖T‖+‖T‖2

| Im λ |(1−‖T‖2).

The estimate (4.6) can be derived similarly by using the second Schur complement. �

In the following we are going to show that Theorem 4.1 is just a consequence of Theorem 4.3.However, since the enclosure in Theorem 4.1 is expressed in terms of the spectral quantities of Aand D, compared with Theorem 4.3 it gives a more intuitive and explicit insight into the locationof the spectrum of S.

Lemma 4.6. Let c > 0. Then there exists C > 0 (depending on c) such that∣∣x1+1/n− x∣∣≤ C

nfor all x ∈ [0,c] and all n ∈ N.


Proof. Let x ∈ (0,c]. By the mean value theorem applied to the function t 7→ xt there exists aξ ∈ (1,1+ 1

n) such that

x1+1/n− x =1n

xξ logx.

If x≤ 1, then ∣∣x1+1/n− x∣∣≤ 1

nx| logx| ≤ 1

en.

If c > 1 and x ∈ (1,c], then∣∣x1+1/n− x∣∣≤ 1

nx1+1/n logx≤ 1

nx2 logx≤ c2 logc

n.

This proves the lemma. �

Let H1 and H2 be Hilbert spaces, T a self-adjoint operator in H1 and V ∈ L(H2,H1). Thenby σV (T ) we denote the support of the positive operator-valued measure V ∗ET (·)V , where ETstands for the spectral measure of T . Clearly, σV (T ) is a closed subset of σ(T ). It is compact ifand only if ran V ⊆ ET (∆)H for some bounded set ∆⊆ R.

Proposition 4.7. Let T be a self-adjoint operator in H1 and V ∈ L(H2,H1). Then for λ ∈C\Rthe following statements are equivalent:

(a) ‖(T −λ )−1V‖ ≥ 1;(b) for all f ∈C+(σ(T )) we have λ ∈

⋃t∈σV (T ) B f (t)−1‖ f (T )V‖(t).

Proof. (b)⇒ (a). Let f (t) := |t−λ |−1, t ∈ σ(T ). Then f ∈C+(σ(T )) and thus, by (b), we haveλ ∈ B f (t0)−1‖ f (T )V‖(t0) for some t0 ∈ σV (T ). This means that

|t0−λ | ≤ f (t0)−1‖ f (T )V‖= |t0−λ | · ‖(T −λ )−1V‖,

which is (a).(a)⇒ (b). Let f ∈ C+(σ(T )). It is obvious that f can be extended to a function in C+(R).

Choose such an extension and also denote it by f . For n ∈ N we set

gn(t) := f (t)|t−λ |1+1/n, t ∈ R.

Then each gn is continuous and positive. Note that for |t| ≥ 2|µ|, where µ = Re λ , we have|t − λ | ≥ |t − µ| ≥ |t| − |µ| ≥ 1

2 |t| and hence |t − λ | f (t) ≥ 12 |t| f (t), which is bounded below

by a positive constant. This implies that lim|t|→∞ |gn(t)| = ∞. Thus, for each n ∈ N there existstn ∈ σV (T ) such that |gn(tn)|= δ (gn) := inft∈σV (T ) |gn(t)|.

On the other hand, dom |T −λ |1+1/n ⊆ dom gn(T ) and f (T ) = gn(T )|T −λ |−1−1/n ∈ L(H1).For arbitrary h ∈H2 with ‖h‖ = 1 define the positive measure µh := ‖ET (·)V h‖2, which hassupport contained in σV (T ). Then,

‖ f (T )V h‖2 = ‖gn(T )|T −λ |−1−1/nV h‖2 =∫

σV (T )

|gn(t)|2

|t−λ |2+2/n dµh(t)

≥ δ2(gn)

∥∥|T −λ |−1−1/nV h∥∥2,

and hence

‖ f (T )V‖ ≥ δ (gn)∥∥|T −λ |−1−1/nV

∥∥= |gn(tn)|∥∥|T −λ |−1−1/nV

∥∥= |tn−λ | f (tn) · |tn−λ |1/n

∥∥|T −λ |−1−1/nV∥∥. (4.12)


Now, consider the functions hn(t) := |t − λ |−1−1/n, n ∈ N, and h(t) := |t − λ |−1. Since |t −λ |−1 ≤ | Im λ |−1 for all t ∈ R, it follows from Lemma 4.6 that there exists C > 0 such that|hn(t)−h(t)| ≤C/n for all t ∈ R. This, together with (a), implies that

1−∥∥|T −λ |−1−1/nV

∥∥≤ ∥∥(T −λ )−1V∥∥−∥∥|T −λ |−1−1/nV

∥∥= ‖h(T )V‖−‖hn(T )V‖ ≤

∥∥h(T )V −hn(T )V∥∥≤ C‖V‖

n,

which, in turn, yields ∥∥|T −λ |−1−1/nV∥∥n ≥

(1−C‖V‖

n

)n

.

As the right-hand side tends to e−C‖V‖ as n→∞, there exists γ > 0 such that∥∥|T−λ |−1−1/nV

∥∥≥γ1/n for all n ∈N. Hence, if there exists some n ∈N such that |tn−λ | ≥ 1/γ , we find from (4.12)that ‖ f (T )V‖ ≥ |tn−λ | f (tn), which means that λ ∈ B f (tn)−1‖ f (T )V‖(tn). Otherwise, there existsa subsequence (tnk) such that tnk → t0 as k→ ∞ with t0 ∈ σV (T ). In this case, replacing n by nkin (4.12) and letting k→ ∞ we obtain

‖ f (T )V‖ ≥ |t0−λ | f (t0) · ‖(T −λ )−1V‖ ≥ |t0−λ | f (t0),

that is, λ ∈ B f (t0)−1‖ f (T )V‖(t0). �

Proposition 4.7 and Theorem 4.3 now immediately imply the following slight improvement ofTheorem 4.1.

Theorem 4.8. Let S be the block operator matrix in (4.1). Then, for any f ∈ C+(σ(A)) andg ∈C+(σ(D)) we have

σ(S)\R⊆

( ⋃t∈σB(A)

B f (t)−1‖ f (A)B‖(t)

)∩

( ⋃s∈σB∗ (D)

Bg(s)−1‖g(D)B∗‖(s)

). (4.13)

We shall now check the performance of several spectral enclosures for block operator matricesfrom above and from the literature on a specific example. The result is illustrated in Figure 7below.

Example 4.9. Let H+ = H− = C2, consider the matrices

A =

[1 00 2

], B =

[13 00 2

3

], D =

[1 00 1

],

and let S be as in (4.1). The eigenvalues of S are given by

1± 13

i and32±√

76

i. (4.14)

1. The spectral enclosure from [25, Theorem 2.7] states that

σ(S)\R⊆ {λ ∈ ρ(A) : ‖(A−λ )−1‖−1 ≤ ‖B‖} ∪ {λ ∈ ρ(D) : ‖(D−λ )−1‖−1 ≤ ‖B‖}

= B 23(1)∪B 2

3(2).

2. The enclosure in [5, Theorem 3.5] (see also (4.4)) yields the estimate

σ(S)\R⊆ (B 23(1)∪B 2

3(2))∩B 2

3(1) = B 2

3(1).


3. The next enclosure that we check is (4.8). Since all matrices are diagonal, we have N(λ ) =

N(λ ) = M(λ ) = M(λ ). Thus (4.8) is

σ(S)\R ⊆{

λ ∈ C\R : ‖B(D−λ )−1B∗(A−λ )−1‖ ≥ 1}. (4.15)

We have

B(D−λ )−1B∗(A−λ )−1 =19

[(1−λ )−2 0

0 4(1−λ )−1(2−λ )−1

]and hence

‖B(D−λ )−1B∗(A−λ )−1‖= 19

max{

1|λ −1|2

,4

|λ −1| |λ −2|

}.

Therefore a non-real complex number λ is in the right-hand side of (4.15) if and only if

9|λ −1|2 ≤ 1 or 9|λ −1| |λ −2| ≤ 4.

Since the first inequality implies the second, we obtain that (4.15) is equivalent to

σ(S)\R⊆{

λ ∈ C\R : |λ −1| |λ −2| ≤ 49

}. (4.16)

4. To compute (4.5) in Theorem 4.3, we observe that

‖(A−λ )−1B‖= 13

max{|1−λ |−1,2|2−λ |−1} and ‖(D−λ )−1B∗‖= 23|1−λ |

.

This leads to

σ(S)\R ⊆ (B 13(1)∪B 2

3(2))∩B 2

3(1) = B 1

3(1)∪ (B 2

3(1)∩B 2

3(2)). (4.17)

5. Let us now discuss our spectral enclosure from Theorem 4.1. Choose g(t) = f (t) = |t|−1,which is valid since A and D are invertible. Then

‖ f (A)B‖= ‖A−1B‖= 13

and ‖g(D)B∗‖= ‖D−1B∗‖= 23.

Hence, (4.3) yields

σ(S)\R⊆(B 1

3(1)∪B 2

3(2))∩B 2

3(1) = B 1

3(1)∪

(B 2

3(1)∩B 2

3(2)), (4.18)

which is the same as (4.17).The right-hand side of (4.17) (or (4.18)) is obviously contained in the right-hand side of

(4.16); actually, it is significantly smaller (e.g. the interval (53 ,

73) is in the right-hand side of

(4.16) but not in the right-hand side of (4.18)). Note that the first three estimates have theeigenvalues 1± 1

3 i in their interior, while all four eigenvalues of S lie on the boundary of theregion given in (4.17).

In the following corollary we consider a useful special case of Theorem 4.1. We denote byC+ and C− the open right and left half-planes, respectively. The enclosure described in Corol-lary 4.10 is illustrated in Figure 1.

Corollary 4.10. Let S be the block operator matrix in (4.1) and assume that 0 ∈ ρ(A). Then

σ(S)\R⊆⋃

a∈σB(A)

B|a|‖A−1B‖(a). (4.19)


1 2

FIGURE 7. The spectral enclosure for σ(S) \R in (4.17) for the operator inExample 4.9 is a union of a disc and the intersection of two discs (filled orangeregion). The boundary of the set on the right-hand side of (4.16) is the bluedashed line. The eigenvalues of S (see (4.14)) are indicated with black dots.

Assume, in addition, that ‖A−1B‖< 1. Then the right-hand side of (4.19) is contained in the set{z ∈ C : | Im z| ≤ ‖A−1B‖√

1−‖A−1B‖2|Re z|

}, (4.20)

which is a double-sector with half opening angle arcsin‖A−1B‖; moreover,

(σ(S)\R

)∩C+ ⊆

{{z ∈ C : Re z≥

(1−‖A−1B‖

)min(σ(A)∩ (0,∞)

)}, σ(A)∩ (0,∞) 6=∅

∅, otherwise,

and(σ(S)\R

)∩C−⊆

{{z ∈ C : Re z≤

(1−‖A−1B‖

)max

(σ(A)∩ (−∞,0)

)}, σ(A)∩ (−∞,0) 6=∅

∅, otherwise.

Proof. Since 0 ∈ ρ(A), the inclusion (4.19) follows from Theorem 4.1 by setting f (t) = |t|−1.Now assume also that ‖A−1B‖< 1. It is elementary to check that the lines

Im z =± ‖A−1B‖√1−‖A−1B‖2

Re z

touch the discs B|a|‖A−1B‖(a), a ∈ σ(A), tangentially. Further, these discs are contained in thedouble sector enclosed by the two lines (see (4.20)). Hence, the right-hand side of (4.19) iscontained in (4.20).

Finally, if σ(A)∩ (0,∞) 6=∅, then, for every a ∈ σ(A)∩ (0,∞) and z ∈ Ba‖A−1B‖(a), we have

Re z≥ a−a‖A−1B‖ ≥(1−‖A−1B‖

)min(σ(A)∩ (0,∞)

).

Similarly, if σ(A)∩ (−∞,0) 6= ∅, then, for every a ∈ σ(A)∩ (−∞,0) and z ∈ Ba‖A−1B‖(a), wehave

Re z≤(1−‖A−1B‖

)min(σ(A)∩ (−∞,0)

).

This shows the inclusions for (σ(S)\R)∩C+ and (σ(S)\R)∩C−. �

A similar result holds when one replaces A and B by D and B∗, respectively. More precisely,if 0 ∈ ρ(D) then

σ(S)\R⊆⋃

d∈σB∗ (D)

B|d|‖D−1B∗‖(d). (4.21)


If it is also assumed that ‖D−1B∗‖< 1, then

σ(S)\R⊆

{z ∈ C : | Im z| ≤ ‖D−1B∗‖√

1−‖D−1B∗‖2|Re z|

}, (4.22)

which is a double-sector with half opening angle arcsin‖D−1B∗‖; moreover,

(σ(S)\R

)∩C+⊆

{{z ∈ C : Re z≥

(1−‖D−1B∗‖

)min(σ(D)∩ (0,∞)

)}, σ(D)∩ (0,∞) 6=∅

∅, otherwise,

and(σ(S)\R

)∩C−⊆

{{z ∈ C : Re z≤

(1−‖D−1B∗‖

)max

(σ(D)∩ (−∞,0)

)}, σ(D)∩ (−∞,0) 6=∅

∅, otherwise.

5. Application to J-frame operators

Originally, frame theory has been developed for Hilbert spaces; see, e.g. [7] and the referencestherein. A frame for a Hilbert space (H ,(., .)) is a family of vectors F = { fi}i∈I for which thereexist constants 0 < α ≤ β < ∞ such that

α ‖ f‖2 ≤∑i∈I|〈 f , fi〉|2 ≤ β ‖ f‖2, for every f ∈H . (5.1)

The optimal constants α and β for which (5.1) holds are known as the frame bounds of F .Recently, various approaches have been suggested to introduce frame theory also to Krein

spaces; see [9, 12, 24]. In this section we apply our results to J-frame operators as introduced in[12]; see also [11]. In particular, we improve the enclosure for the non-real spectrum of J-frameoperators obtained in [11]; see Theorem 5.2 below.

An indefinite inner product space (H , [· , ·]) is a (complex) vector space H endowed with aHermitian sesquilinear form [., .]. Given a subspace S of H , the orthogonal subspace to S isdefined by

S [⊥] = {x ∈H : [x,s] = 0 for every s ∈S },and S is called non-degenerate if S ∩S [⊥] = {0}. If S and T are subspaces of H , thenotation S [⊥]T stands for S ⊆T [⊥].

A Krein space is a non-degenerate indefinite inner product space (H , [· , ·]) which admits adecomposition H =H+ uH− such that H+ [⊥]H− and (H±,±[· , ·]) are Hilbert spaces. Sucha decomposition is often called a fundamental decomposition and it is denoted H =H+ [u]H−.

The Hilbert spaces (H±,±[· , ·]) induce in a natural way a positive definite inner product (., .)on H such that (H ,(., .)) is a Hilbert space. Observe that the inner products [., .] and (., .) of Hare related by means of a fundamental symmetry, i.e. a unitary self-adjoint operator J ∈ L(H )that satisfies

( f ,g) = [J f ,g] , f ,g ∈H .

Although the fundamental decomposition is not unique, the norms induced by different funda-mental decompositions turn out to be equivalent; see, e.g. [16, Proposition I.1.2]. Therefore, the(Hilbert space) topology in H does not depend on the chosen fundamental decomposition.

Let us now introduce J-frames. Given a Krein space (H , [., .]), consider a frame F = { fi}i∈Ifor the associated Hilbert space (H ,(., .)) and set

I+ := {i ∈ I : [ fi, fi]≥ 0} and I− := {i ∈ I : [ fi, fi]< 0}.


Then F is called a J-frame for H if M+ := span{ fi : i ∈ I+} and M− := span{ fi : i ∈ I−} arenon-degenerate subspaces of H and there exist constants 0 < α± ≤ β± such that

α±(±[ f , f ])≤ ∑i∈I±

∣∣[ f , fi]∣∣2 ≤ β±(±[ f , f ]) for f ∈M±; (5.2)

see [12, Theorem 3.9]. The spaces (M±,±[· , ·]) are then Hilbert spaces by [12, Proposition 3.8]and the optimal constants 0 < α± ≤ β± are called the J-frame bounds of F .

Note that (5.2) says that F+ = { fi}i∈I+ and F− = { fi}i∈I− are frames for the Hilbert spaces(M+, [., .]) and (M−,−[., .]), respectively. Moreover, the frame bounds for F+ and F− areα+,β+ and α−,β−, respectively. Also note that not necessarily M+ [⊥]M−.

The J-frame operator associated with F is defined by

S f = ∑i∈I+

[ f , fi] fi− ∑i∈I−

[ f , fi] fi, f ∈H .

It plays a fundamental role in the indefinite reconstruction formula (see [12]). The operator S isan invertible, bounded, self-adjoint operator in the Krein space H . The following representationfor J-frame operators was obtained in [11, Theorems 3.1 and 3.2].

Theorem 5.1. Given a bounded self-adjoint operator S in a Krein space (H , [., .]), the followingconditions are equivalent.

(i) S is a J-frame operator.(ii) There exists a fundamental decomposition

H = H+[.+]H− (5.3)

such that S admits a representation with respect to (5.3) of the form

S =

[A −AK

K∗A D

](5.4)

where A is a uniformly positive operator in the Hilbert space (H+, [., .]), K : H−→H+ isa uniform contraction1 (i.e. ‖K‖< 1), and D is a self-adjoint operator such that D+K∗AKis uniformly positive in the Hilbert space (H−,−[., .]).

(iii) There exists a fundamental decomposition

H = K+[.+]K− (5.5)

such that S admits a representation with respect to (5.5) of the form

S =

[A′ LD′

−D′L∗ D′

](5.6)

where D′ is a uniformly positive operator in (K−,−[., .]), L : K− → K+ is a uniformcontraction1, and A′ is a self-adjoint operator such that A′+ LD′L∗ is uniformly positivein (K+, [., .]).

The representations for the J-frame operator given in Theorem 5.1 were used to show thatthe J-frame bounds for F are related to the boundary of the spectrum of the uniformly positiveoperators D + K∗AK and A′ + LD′L∗. More precisely, [11, Proposition 4.1] says that if S isrepresented as in (5.4), then

α− = min σ(D+K∗AK) and β− = max σ(D+K∗AK). (5.7)

1The operator norm used depends on the norm induced by the respective fundamental decomposition (5.3) or (5.5).


On the other hand, if S is represented as in (5.6), then

α+ = min σ(A′+LD′L∗) and β+ = max σ(A′+LD′L∗). (5.8)

Given a J-frame F = { fi}i∈I for H with J-frame operator S, the canonical dual J-frame ofF is defined as F ′ = {S−1 fi}i∈I . It is also a J-frame for H such that F ′

± = {S−1 fi}i∈I± areframes for (M [⊥]

∓ ,±[., .]), i.e. there exist constants 0 < γ± ≤ δ± such that

γ±(±[ f , f ])≤ ∑i∈I±

∣∣[ f ,S−1 fi]∣∣2 ≤ δ±(±[ f , f ]) for every f ∈M

[⊥]∓ .

The J-frame bounds of F ′ are also related to the representations in Theorem 5.1: if S is repre-sented as in (5.6) then

γ− = min σ((D′)−1)= (max σ(D′)

)−1 and δ− = max σ((D′)−1)= (min σ(D′)

)−1, (5.9)

and if S is represented as in (5.4) then

γ+ = min σ(A−1) =(max σ(A)

)−1 and δ+ = max σ(A−1) =(min σ(A)

)−1; (5.10)

see [11, Proposition 4.2].The following theorem gives an enclosure for the non-real spectrum of the J-frame operator S

of a J-frame F in terms of the J-frame bounds associated with F and its canonical dual J-frameF ′.

Theorem 5.2. Let F be a J-frame for (H , [., .]) with J-frame operator S. Then,

σ(S)\R ⊆

⋃a∈[δ−1

+ ,γ−1+ ]

Ba‖K‖(a)

∩ ⋃

b∈[α−,β−]B b‖K‖

1−‖K‖2

(b

1−‖K‖2

) , (5.11)

where K is the angular operator appearing in (5.4). Also,

σ(S)\R ⊆

⋃d∈[δ−1

− ,γ−1− ]

Bd‖L‖(d)

∩ ⋃

b∈[α+,β+]

B b‖L‖1−‖L‖2

(b

1−‖L‖2

) , (5.12)

where L is the angular operator appearing in (5.6). The sets on the right-hand sides of (5.11)and (5.12) are contained in sectors of the form {z ∈C+ : | Im z| ≤ tanϕ ·Re z} with half openingangles ϕ = arcsin‖K‖ and ϕ = arcsin‖L‖, respectively.

Proof. Let σB(A) be defined as in Theorem 4.1. Obviously we have σB(A) ⊆ σ(A) ⊆ [a−,a+],where the constants a± are defined in (3.1). Applying Corollary 4.10 to S represented as in (5.4)we obtain that

σ(S)\R ⊆⋃

a∈[a−,a+]Ba‖K‖(a). (5.13)

Moreover, according to (5.10) we have that a− = δ−1+ and a+ = γ

−1+ . On the other hand, S−1 is

also a J-frame operator and it is easy to check that

S−1 =

[A−1−KZK∗ KZ−ZK∗ Z

],

where Z := (D+K∗AK)−1 is a uniformly positive operator; cf. Theorem 5.1. Therefore Corol-lary 4.10 applied to S−1 represented as above implies

σ(S−1)\R ⊆⋃

r∈[r−,r+]Br‖K∗‖ (r) ,


where [r−,r+] is the closure of the numerical range of Z. Also, (5.7) says that r− = β−1− and

r+ = α−1− . With b := 1

r it follows that

σ(S−1)\R ⊆⋃

b∈[α−,β−]B ‖K‖

b

( 1b

).

Recall that λ ∈ σ(S)\{0} if and only if 1λ∈ σ(S−1)\{0}. Moreover, observe that for r > 0

1λ∈ B‖K‖

r

( 1r

)if and only if λ ∈ B r‖K‖

1−‖K‖2

(r

1−‖K‖2

).

Therefore,σ(S)\R ⊆

⋃b∈[α−,β−]

B b‖K‖1−‖K‖2

(b

1−‖K‖2

), (5.14)

and (5.11) follows by intersecting (5.13) and (5.14).The proof of (5.12) is similar. It follows from Corollary 4.10 applied to S represented as in

(5.6), and also to S−1 represented as

S−1 =

[Z′ −Z′L

L∗Z′ (D′)−1−L∗Z′L

],

with Z′ = (A′+LD′L∗)−1.The statement about the sectors is clear from Corollary 4.10. �

In the following, we compare Theorem 5.2 with the enclosure for the non-real spectrum ofJ-frame operators obtained in [11].

Let F be a J-frame for a Krein space (H , [., .]) with J-frame operator S and J-frame bounds0<α±≤ β±. Assume also that 0< γ±≤ δ± are the J-frame bounds of its canonical dual J-frameF ′. In [11, Corollary 5.3] it was shown that

σ(S)\R ⊆ Bmin{γ−1+ ,γ−1

− }(min{γ−1

+ ,γ−1− })∩{

λ ∈ C : Re λ ≥ max{α+,α−}2

}. (5.15)

Here, Br(a) denotes the interior of Br(a). Let us show that the intersection of the sets on theright-hand sides of (5.11) and (5.12) are (strictly) contained in the right-hand side of (5.15).

For every a ∈ [δ−1+ ,γ−1

+ ] it is easy to see that Ba‖K‖(a) is strictly contained in Bγ−1+(γ−1

+ ).Therefore, ⋃

a∈[δ−1+ ,γ−1

+ ]

Ba‖K‖(a)⊆ Bγ−1+(γ−1

+ ).

On the other hand, given r > 0, if λ ∈ B r‖K‖1−‖K‖2

(r

1−‖K‖2

), then Re λ ≥ r

1+‖K‖ >r2 . Thus,

⋃b∈[α−,β−]

B b‖K‖1−‖K‖2

(b

1−‖K‖2

)⊆{

λ ∈ C : Re λ ≥ α−2

}.

Similarly, it is easy to see that ⋃d∈[δ−1

− ,γ−1− ]

Bd‖L‖(d)⊆ Bγ−1−(γ−1− ),

and ⋃b∈[α+,β+]

B b‖L‖1−‖L‖2

(b

1−‖L‖2

)⊆ {λ ∈ C : Re λ ≥ α+

2

}.


Hence, Theorem 5.2 improves the enclosure (5.15) for the non-real spectrum of the J-frameoperator S obtained in [11].

References[1] A. Adamjan and H. Langer, Spectral properties of a class of rational operator valued functions, J. Operator

Theory 33 (1995), 259–277.[2] S. Albeverio and A. K. Motovilov, On invariant graph subspaces of a J-self-adjoint operator in the Feshbach

case, Math. Notes 100 (2016), 761–773.[3] S. Albeverio, A. K. Motovilov and A. A. Shkalikov, Bounds on variation of spectral subspaces under J-self-

adjoint perturbations, Integral Equations Operator Theory 64 (2009), 455–486.[4] S. Albeverio, A. K. Motovilov and C. Tretter, Bounds on the spectrum and reducing subspaces of a J-self-

adjoint operator, Indiana Univ. Math. J. 59 (2010), 1737–1776.[5] J. Behrndt, F. Philipp and C. Trunk, Bounds on the non-real spectrum of differential operators with indefinite

weights, Math. Ann. 357 (2013), 185–213.[6] E.P. Bogomolova, Some questions of the spectral analysis of a nonselfadjoint differential operator with a ”float-

ing” singularity in the coefficient, Differential Equations 21 (1985), 1229–1234.[7] O. Christensen, An Introduction to Frames and Riesz Bases, Birkhauser, Boston, 2003.[8] A. Dall’Aqua, D. Mugnolo and M. Schelling, A new Gershgorin-type result for the localisation of the spectrum

of matrices, Math. Nach. 288 (2015), 1981–1994.[9] K. Esmeral, O. Ferrer and E. Wagner, Frames in Krein spaces arising from a non-regular W -metric, Banach J.

Math. Anal. 9 (2015), 1–16.[10] S. Gershgorin, Uber die Abgrenzung der Eigenwerte einer Matrix, Bull. Acad. des Sci. URSS 6 (1931), 749–754

[German].[11] J. I. Giribet, M. Langer, L. Leben, A. Maestripieri, F. Martınez Perıa and C. Trunk, Spectrum of J-frame

operators, Opuscula Math. 38 (2018), 623–649.[12] J. I. Giribet, A. Maestripieri, F. Martınez Perıa and P. G. Massey, On frames for Krein spaces, J. Math. Anal.

Appl. 393 (2012), 122–137.[13] P. Jonas and C. Trunk, On a class of analytic operator functions and their linearizations, Math. Nachr. 243

(2002), 92–133.[14] P. Jonas and C. Trunk, A Sturm-Liouville problem depending rationally on the eigenvalue parameter, Math.

Nachr. 280 (2007), 1709-1726.[15] T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin–Heidelberg, 1995.[16] H. Langer, Spectral functions of definitizable operators in Krein spaces, in: Functional Analysis. (Proceedings

of a Conference held at Dubrovnik, 1981.) Lecture Notes in Math., vol. 948, Springer, Berlin–New York, 1982,pp. 1–46.

[17] H. Langer, M. Langer, A. Markus and C. Tretter, Spectrum of definite type of self-adjoint operators in Kreinspaces, Linear Multilinear Algebra 53 (2005), 115–136.

[18] H. Langer, M. Langer and C. Tretter, Variational principles for eigenvalues of block operator matrices, IndianaUniv. Math. J. 51 (2002), 1427–1459.

[19] H. Langer, A. Markus and V. Matsaev, Locally definite operators in indefinite inner product spaces, Math. Ann.308 (1997), 405–424.

[20] H. Langer, R. Mennicken and M. Moller, A second order differential operator depending non-linearly on theeigenvalue parameter, in: Topics in Operator Theory. Ernst D. Hellinger Memorial Volume. Oper. Theory Adv.Appl., vol. 48, Birkhauser, Basel, 1990, pp. 319–332.

[21] H. Langer and C. Tretter, Spectral decomposition of some nonselfadjoint block operator matrices, J. OperatorTheory 39 (1998), 339–359.

[22] M. Langer and M. Strauss, Spectral properties of unbounded J-self-adjoint block operator matrices, J. Spectr.Theory 7 (2017), 137–190.

[23] R. Mennicken and A. A. Shkalikov, Spectral decomposition of symmetric operator matrices, Math. Nachr. 179(1996), 259–273.

[24] I. Peng and S. Waldron, Signed frames and Hadamard products of Gram matrices, Linear Algebra Appl. 347(2002), 131–157.

[25] H. N. Salas, Gershgorin’s theorem for matrices of operators, Lineal Algebra Appl. 291 (1999), 15–36.[26] C. Tretter, Spectral Theory of Block Operator Matrices and Applications, Imperial College Press, London,

2008.[27] C. Tretter, Spectral inclusion for unbounded block operator matrices, J. Funct. Anal. 256 (2009), 3806–3829.


DEPARTAMENTO DE MATEMATICA – FI-UBA AND INSTITUTO ARGENTINO DE MATEMATICA “ALBERTO P.CALDERON” (CONICET), SAAVEDRA 15 (1083) BUENOS AIRES, ARGENTINA

Email address: [email protected]

DEPARTMENT OF MATHEMATICS AND STATISTICS, UNIVERSITY OF STRATHCLYDE, 26 RICHMOND STREET,GLASGOW G1 1XH, UNITED KINGDOM

Email address: [email protected]: personal.strath.ac.uk/m.langer

CENTRO DE MATEMATICA DE LA PLATA – FACULTAD DE CIENCIAS EXACTAS, UNIVERSIDAD NACIONAL

DE LA PLATA, C.C. 172, (1900) LA PLATA, ARGENTINA AND INSTITUTO ARGENTINO DE MATEMATICA “AL-BERTO P. CALDERON” (CONICET), SAAVEDRA 15 (1083) BUENOS AIRES, ARGENTINA

Email address: [email protected]

KATHOLISCHE UNIVERSITAT EICHSTATT-INGOLSTADT, OSTENSTRASSE 26, 85072 EICHSTATT, GERMANY

Email address: [email protected]: www.ku.de/?fmphilipp

INSTITUTO ARGENTINO DE MATEMATICA “ALBERTO P. CALDERON” (CONICET), SAAVEDRA 15 (1083)BUENOS AIRES, ARGENTINA AND INSTITUT FUR MATHEMATIK, TECHNISCHE UNIVERSITAT ILMENAU, POST-FACH 100565, D-98684 ILMENAU, GERMANY

Email address: [email protected]: www.tu-ilmenau.de/de/analysis/team/carsten-trunk

Documents

Spectral enclosures for a class of block operator matrices filespectral enclosures for a class of block operator matrices juan giribet, matthias langer, francisco mart´inez per ´ia,